answer:The first step is to find the derivative of f(x), which is f'(x) = -1331x^2. At x = 3, f'(3) = -1331 cdot 3^2 = -3993. The inverse function series is given by f^{-1}(x) = a + frac{x - f(a)}{f'(a)}, where a is the center of the series, which is 3 in this case. We substitute f(a) and f'(a): f(3) = -frac{1331 cdot 3^3}{27} = -1331. Now, we plug these values into the formula: f^{-1}(x) = 3 + frac{x + 1331}{-3993}. Simplifying, we get the first-order series of the inverse function: f^{-1}(x) = frac{-x - 1331}{3993} + 3.

answer:The roots of the polynomial f(x) can be found using the quadratic formula, x = frac{-b pm sqrt{b^2 - 4ac}}{2a}. For the given polynomial f(x) = -frac{27x^2}{2} + frac{x}{2} - frac{7}{4}, we have a = -frac{27}{2}, b = frac{1}{2}, and c = -frac{7}{4}. Applying the formula: x = frac{-left(frac{1}{2}right) pm sqrt{left(frac{1}{2}right)^2 - 4left(-frac{27}{2}right)left(-frac{7}{4}right)}}{2left(-frac{27}{2}right)} x = frac{-frac{1}{2} pm sqrt{frac{1}{4} - frac{189}{4}}}{-27} x = frac{-frac{1}{2} pm sqrt{-47}}{-27} Since we have a negative number under the square root, the roots will be complex: x = frac{-frac{1}{2} pm isqrt{47}}{-27} x = frac{1}{54} left(-1 pm isqrt{47}right) Hence, the roots of the polynomial are: x = frac{1}{54} left(-1 - isqrt{47}right) quad text{and} quad x = frac{1}{54} left(-1 + isqrt{47}right)

answer:The trigonal planar molecular geometry is characterized by a central atom with three covalently bonded atoms and no non-bonding valence electron pairs. In the given options, we are looking for a carbon atom that forms a double bond with one atom and single bonds with two others, maintaining a total of four bonds. Let's analyze each option: a) CH₃CHO (formaldehyde) contains a carbonyl group (C=O), where the carbon atom is double-bonded to an oxygen atom and single-bonded to two hydrogen atoms, fulfilling the criteria for trigonal planar geometry. b) CO₂ (carbon dioxide) has a carbon atom double-bonded to two oxygen atoms. This results in a linear geometry, not trigonal planar. c) CH₃Cl (chloromethane) has a carbon atom bonded to one chlorine atom and three hydrogen atoms, resulting in a tetrahedral geometry. Thus, the correct answer is: a) CH₃CHO

answer:The simplified form of the expression is frac{279841}{256} left(cos left(frac{2 pi }{15}right)-i sin left(frac{2 pi }{15}right)right). This result is obtained by first raising the complex number inside the parentheses to the power of 4, then simplifying the trigonometric functions and their product. The De Moivre's theorem is used in the simplification process, which states that for any real number theta and integer n, (cos theta - i sin theta)^n = cos (ntheta) - i sin (ntheta).